why is my image texture with texture not displaying properly?
To make our life easy blender can remember an image that you choose in the UV/mage editor while uv unwrapping or that you create while texture painting and use it when drawing an object in the 3d viewport. This only happens in the viewport with the right settings but does not carry over into the final render, to use an image texture in the final render you need to setup your material properly.

Recent versions of blender added the ability to add images to paint onto, when using blender internal the texture is normally also created, in cycles when needed a material and/or an image texture node can be created but it doesn't get connected up, you need to do that yourself.

If using blender internal you need to ensure that the texture is added and setup to use the image. For cycles you need to ensure the image texture node is connected to the shader node that you want to use.